Theorem opphllem1 25639

Description: Lemma for opphl 25646. (Contributed by Thierry Arnoux, 20-Dec-2019.)

Hypotheses

Ref	Expression
hpg.p	|- P = ( Base ` G )
hpg.d	|- .- = ( dist ` G )
hpg.i	|- I = (Itv ` G )
hpg.o	|- O = { <. a , b >. | ( ( a e. ( P \ D ) /\ b e. ( P \ D ) ) /\ E. t e. D t e. ( a I b ) ) }
opphl.l	|- L = (LineG ` G )
opphl.d	|- ( ph -> D e. ran L )
opphl.g	|- ( ph -> G e. TarskiG )
opphllem1.s	|- S = ( (pInvG ` G ) ` M )
opphllem1.a	|- ( ph -> A e. P )
opphllem1.b	|- ( ph -> B e. P )
opphllem1.c	|- ( ph -> C e. P )
opphllem1.r	|- ( ph -> R e. D )
opphllem1.o	|- ( ph -> A O C )
opphllem1.m	|- ( ph -> M e. D )
opphllem1.n	|- ( ph -> A = ( S ` C ) )
opphllem1.x	|- ( ph -> A =/= R )
opphllem1.y	|- ( ph -> B =/= R )
opphllem1.z	|- ( ph -> B e. ( R I A ) )

Assertion

Ref	Expression
opphllem1	|- ( ph -> B O C )

Distinct variable groups:   D, a, b   I, a, b   P, a, b   t, A   t, B   t, D   t, R   t, C   t, G   t, L   t, I   t, M   t, O   t, P   t, S   ph, t   t, .-   t, a, b

Allowed substitution hints:   ph( a, b)   A( a, b)   B( a, b)   C( a, b)   R( a, b)   S( a, b)   G( a, b)   L( a, b)   M( a, b)   .- ( a, b)   O( a, b)

Proof of Theorem opphllem1

Step	Hyp	Ref	Expression
1		simpr 477	. . . . . 6 |- ( ( ( ph /\ B e. D ) /\ A = B ) -> A = B )
2		simplr 792	. . . . . 6 |- ( ( ( ph /\ B e. D ) /\ A = B ) -> B e. D )
3	1, 2	eqeltrd 2701	. . . . 5 |- ( ( ( ph /\ B e. D ) /\ A = B ) -> A e. D )
4		hpg.p	. . . . . . 7 |- P = ( Base ` G )
5		hpg.i	. . . . . . 7 |- I = (Itv ` G )
6		opphl.l	. . . . . . 7 |- L = (LineG ` G )
7		opphl.g	. . . . . . . 8 |- ( ph -> G e. TarskiG )
8	7	ad2antrr 762	. . . . . . 7 |- ( ( ( ph /\ B e. D ) /\ A =/= B ) -> G e. TarskiG )
9		opphllem1.b	. . . . . . . 8 |- ( ph -> B e. P )
10	9	ad2antrr 762	. . . . . . 7 |- ( ( ( ph /\ B e. D ) /\ A =/= B ) -> B e. P )
11		opphl.d	. . . . . . . . 9 |- ( ph -> D e. ran L )
12		opphllem1.r	. . . . . . . . 9 |- ( ph -> R e. D )
13	4, 6, 5, 7, 11, 12	tglnpt 25444	. . . . . . . 8 |- ( ph -> R e. P )
14	13	ad2antrr 762	. . . . . . 7 |- ( ( ( ph /\ B e. D ) /\ A =/= B ) -> R e. P )
15		opphllem1.a	. . . . . . . 8 |- ( ph -> A e. P )
16	15	ad2antrr 762	. . . . . . 7 |- ( ( ( ph /\ B e. D ) /\ A =/= B ) -> A e. P )
17		opphllem1.y	. . . . . . . 8 |- ( ph -> B =/= R )
18	17	ad2antrr 762	. . . . . . 7 |- ( ( ( ph /\ B e. D ) /\ A =/= B ) -> B =/= R )
19	18	necomd 2849	. . . . . . . 8 |- ( ( ( ph /\ B e. D ) /\ A =/= B ) -> R =/= B )
20		opphllem1.z	. . . . . . . . 9 |- ( ph -> B e. ( R I A ) )
21	20	ad2antrr 762	. . . . . . . 8 |- ( ( ( ph /\ B e. D ) /\ A =/= B ) -> B e. ( R I A ) )
22	4, 5, 6, 8, 14, 10, 16, 19, 21	btwnlng3 25516	. . . . . . 7 |- ( ( ( ph /\ B e. D ) /\ A =/= B ) -> A e. ( R L B ) )
23	4, 5, 6, 8, 10, 14, 16, 18, 22	lncom 25517	. . . . . 6 |- ( ( ( ph /\ B e. D ) /\ A =/= B ) -> A e. ( B L R ) )
24	11	ad2antrr 762	. . . . . . 7 |- ( ( ( ph /\ B e. D ) /\ A =/= B ) -> D e. ran L )
25		simplr 792	. . . . . . 7 |- ( ( ( ph /\ B e. D ) /\ A =/= B ) -> B e. D )
26	12	ad2antrr 762	. . . . . . 7 |- ( ( ( ph /\ B e. D ) /\ A =/= B ) -> R e. D )
27	4, 5, 6, 8, 10, 14, 18, 18, 24, 25, 26	tglinethru 25531	. . . . . 6 |- ( ( ( ph /\ B e. D ) /\ A =/= B ) -> D = ( B L R ) )
28	23, 27	eleqtrrd 2704	. . . . 5 |- ( ( ( ph /\ B e. D ) /\ A =/= B ) -> A e. D )
29	3, 28	pm2.61dane 2881	. . . 4 |- ( ( ph /\ B e. D ) -> A e. D )
30		hpg.d	. . . . . 6 |- .- = ( dist ` G )
31		hpg.o	. . . . . 6 |- O = { <. a , b >. | ( ( a e. ( P \ D ) /\ b e. ( P \ D ) ) /\ E. t e. D t e. ( a I b ) ) }
32		opphllem1.c	. . . . . 6 |- ( ph -> C e. P )
33		opphllem1.o	. . . . . 6 |- ( ph -> A O C )
34	4, 30, 5, 31, 6, 11, 7, 15, 32, 33	oppne1 25633	. . . . 5 |- ( ph -> -. A e. D )
35	34	adantr 481	. . . 4 |- ( ( ph /\ B e. D ) -> -. A e. D )
36	29, 35	pm2.65da 600	. . 3 |- ( ph -> -. B e. D )
37	4, 30, 5, 31, 6, 11, 7, 15, 32, 33	oppne2 25634	. . 3 |- ( ph -> -. C e. D )
38		opphllem1.m	. . . . . 6 |- ( ph -> M e. D )
39	4, 6, 5, 7, 11, 38	tglnpt 25444	. . . . 5 |- ( ph -> M e. P )
40		eqid 2622	. . . . . . 7 |- (pInvG ` G ) = (pInvG ` G )
41		opphllem1.s	. . . . . . 7 |- S = ( (pInvG ` G ) ` M )
42	4, 30, 5, 6, 40, 7, 39, 41, 15	mirbtwn 25553	. . . . . 6 |- ( ph -> M e. ( ( S ` A ) I A ) )
43		opphllem1.n	. . . . . . . . 9 |- ( ph -> A = ( S ` C ) )
44	43	eqcomd 2628	. . . . . . . 8 |- ( ph -> ( S ` C ) = A )
45	4, 30, 5, 6, 40, 7, 39, 41, 32, 44	mircom 25558	. . . . . . 7 |- ( ph -> ( S ` A ) = C )
46	45	oveq1d 6665	. . . . . 6 |- ( ph -> ( ( S ` A ) I A ) = ( C I A ) )
47	42, 46	eleqtrd 2703	. . . . 5 |- ( ph -> M e. ( C I A ) )
48	4, 30, 5, 7, 13, 32, 15, 9, 39, 20, 47	axtgpasch 25366	. . . 4 |- ( ph -> E. t e. P ( t e. ( B I C ) /\ t e. ( M I R ) ) )
49	7	ad2antrr 762	. . . . . . . . . 10 |- ( ( ( ph /\ ( t e. P /\ ( t e. ( B I C ) /\ t e. ( M I R ) ) ) ) /\ M = R ) -> G e. TarskiG )
50	13	ad2antrr 762	. . . . . . . . . 10 |- ( ( ( ph /\ ( t e. P /\ ( t e. ( B I C ) /\ t e. ( M I R ) ) ) ) /\ M = R ) -> R e. P )
51		simplrl 800	. . . . . . . . . 10 |- ( ( ( ph /\ ( t e. P /\ ( t e. ( B I C ) /\ t e. ( M I R ) ) ) ) /\ M = R ) -> t e. P )
52		simplrr 801	. . . . . . . . . . . 12 |- ( ( ( ph /\ ( t e. P /\ ( t e. ( B I C ) /\ t e. ( M I R ) ) ) ) /\ M = R ) -> ( t e. ( B I C ) /\ t e. ( M I R ) ) )
53	52	simprd 479	. . . . . . . . . . 11 |- ( ( ( ph /\ ( t e. P /\ ( t e. ( B I C ) /\ t e. ( M I R ) ) ) ) /\ M = R ) -> t e. ( M I R ) )
54		simpr 477	. . . . . . . . . . . 12 |- ( ( ( ph /\ ( t e. P /\ ( t e. ( B I C ) /\ t e. ( M I R ) ) ) ) /\ M = R ) -> M = R )
55	54	oveq1d 6665	. . . . . . . . . . 11 |- ( ( ( ph /\ ( t e. P /\ ( t e. ( B I C ) /\ t e. ( M I R ) ) ) ) /\ M = R ) -> ( M I R ) = ( R I R ) )
56	53, 55	eleqtrd 2703	. . . . . . . . . 10 |- ( ( ( ph /\ ( t e. P /\ ( t e. ( B I C ) /\ t e. ( M I R ) ) ) ) /\ M = R ) -> t e. ( R I R ) )
57	4, 30, 5, 49, 50, 51, 56	axtgbtwnid 25365	. . . . . . . . 9 |- ( ( ( ph /\ ( t e. P /\ ( t e. ( B I C ) /\ t e. ( M I R ) ) ) ) /\ M = R ) -> R = t )
58	12	ad2antrr 762	. . . . . . . . 9 |- ( ( ( ph /\ ( t e. P /\ ( t e. ( B I C ) /\ t e. ( M I R ) ) ) ) /\ M = R ) -> R e. D )
59	57, 58	eqeltrrd 2702	. . . . . . . 8 |- ( ( ( ph /\ ( t e. P /\ ( t e. ( B I C ) /\ t e. ( M I R ) ) ) ) /\ M = R ) -> t e. D )
60	7	adantr 481	. . . . . . . . . . 11 |- ( ( ph /\ M =/= R ) -> G e. TarskiG )
61	60	adantlr 751	. . . . . . . . . 10 |- ( ( ( ph /\ ( t e. P /\ ( t e. ( B I C ) /\ t e. ( M I R ) ) ) ) /\ M =/= R ) -> G e. TarskiG )
62	39	adantr 481	. . . . . . . . . . 11 |- ( ( ph /\ M =/= R ) -> M e. P )
63	62	adantlr 751	. . . . . . . . . 10 |- ( ( ( ph /\ ( t e. P /\ ( t e. ( B I C ) /\ t e. ( M I R ) ) ) ) /\ M =/= R ) -> M e. P )
64	13	adantr 481	. . . . . . . . . . 11 |- ( ( ph /\ M =/= R ) -> R e. P )
65	64	adantlr 751	. . . . . . . . . 10 |- ( ( ( ph /\ ( t e. P /\ ( t e. ( B I C ) /\ t e. ( M I R ) ) ) ) /\ M =/= R ) -> R e. P )
66		simplrl 800	. . . . . . . . . 10 |- ( ( ( ph /\ ( t e. P /\ ( t e. ( B I C ) /\ t e. ( M I R ) ) ) ) /\ M =/= R ) -> t e. P )
67		simpr 477	. . . . . . . . . 10 |- ( ( ( ph /\ ( t e. P /\ ( t e. ( B I C ) /\ t e. ( M I R ) ) ) ) /\ M =/= R ) -> M =/= R )
68		simplrr 801	. . . . . . . . . . 11 |- ( ( ( ph /\ ( t e. P /\ ( t e. ( B I C ) /\ t e. ( M I R ) ) ) ) /\ M =/= R ) -> ( t e. ( B I C ) /\ t e. ( M I R ) ) )
69	68	simprd 479	. . . . . . . . . 10 |- ( ( ( ph /\ ( t e. P /\ ( t e. ( B I C ) /\ t e. ( M I R ) ) ) ) /\ M =/= R ) -> t e. ( M I R ) )
70	4, 5, 6, 61, 63, 65, 66, 67, 69	btwnlng1 25514	. . . . . . . . 9 |- ( ( ( ph /\ ( t e. P /\ ( t e. ( B I C ) /\ t e. ( M I R ) ) ) ) /\ M =/= R ) -> t e. ( M L R ) )
71		simpr 477	. . . . . . . . . . 11 |- ( ( ph /\ M =/= R ) -> M =/= R )
72	11	adantr 481	. . . . . . . . . . 11 |- ( ( ph /\ M =/= R ) -> D e. ran L )
73	38	adantr 481	. . . . . . . . . . 11 |- ( ( ph /\ M =/= R ) -> M e. D )
74	12	adantr 481	. . . . . . . . . . 11 |- ( ( ph /\ M =/= R ) -> R e. D )
75	4, 5, 6, 60, 62, 64, 71, 71, 72, 73, 74	tglinethru 25531	. . . . . . . . . 10 |- ( ( ph /\ M =/= R ) -> D = ( M L R ) )
76	75	adantlr 751	. . . . . . . . 9 |- ( ( ( ph /\ ( t e. P /\ ( t e. ( B I C ) /\ t e. ( M I R ) ) ) ) /\ M =/= R ) -> D = ( M L R ) )
77	70, 76	eleqtrrd 2704	. . . . . . . 8 |- ( ( ( ph /\ ( t e. P /\ ( t e. ( B I C ) /\ t e. ( M I R ) ) ) ) /\ M =/= R ) -> t e. D )
78	59, 77	pm2.61dane 2881	. . . . . . 7 |- ( ( ph /\ ( t e. P /\ ( t e. ( B I C ) /\ t e. ( M I R ) ) ) ) -> t e. D )
79		simprrl 804	. . . . . . 7 |- ( ( ph /\ ( t e. P /\ ( t e. ( B I C ) /\ t e. ( M I R ) ) ) ) -> t e. ( B I C ) )
80	78, 79	jca 554	. . . . . 6 |- ( ( ph /\ ( t e. P /\ ( t e. ( B I C ) /\ t e. ( M I R ) ) ) ) -> ( t e. D /\ t e. ( B I C ) ) )
81	80	ex 450	. . . . 5 |- ( ph -> ( ( t e. P /\ ( t e. ( B I C ) /\ t e. ( M I R ) ) ) -> ( t e. D /\ t e. ( B I C ) ) ) )
82	81	reximdv2 3014	. . . 4 |- ( ph -> ( E. t e. P ( t e. ( B I C ) /\ t e. ( M I R ) ) -> E. t e. D t e. ( B I C ) ) )
83	48, 82	mpd 15	. . 3 |- ( ph -> E. t e. D t e. ( B I C ) )
84	36, 37, 83	jca31 557	. 2 |- ( ph -> ( ( -. B e. D /\ -. C e. D ) /\ E. t e. D t e. ( B I C ) ) )
85	4, 30, 5, 31, 9, 32	islnopp 25631	. 2 |- ( ph -> ( B O C <-> ( ( -. B e. D /\ -. C e. D ) /\ E. t e. D t e. ( B I C ) ) ) )
86	84, 85	mpbird 247	1 |- ( ph -> B O C )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   =/= wne 2794   E.wrex 2913   \ cdif 3571   class class class wbr 4653   {copab 4712   ran crn 5115   ` cfv 5888  (class class class)co 6650   Basecbs 15857   distcds 15950  TarskiGcstrkg 25329  Itvcitv 25335  LineGclng 25336  pInvGcmir 25547

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-er 7742  df-pm 7860  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-card 8765  df-cda 8990  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-2 11079  df-3 11080  df-n0 11293  df-xnn0 11364  df-z 11378  df-uz 11688  df-fz 12327  df-fzo 12466  df-hash 13118  df-word 13299  df-concat 13301  df-s1 13302  df-s2 13593  df-s3 13594  df-trkgc 25347  df-trkgb 25348  df-trkgcb 25349  df-trkg 25352  df-cgrg 25406  df-mir 25548

This theorem is referenced by:  opphllem2  25640